In this (first) paper we attempt to generalize the notion of tensor connectivity, subsequently studying how this property is affected in different tensorial operations. We show that the often implied corollary of the linked diagram theorem, namely individual size-extensivity of arbitrary connected closed diagrams, can be violated in Coulomb systems. In particular, the assumption of the existence of localized Hartree–Fock orbitals is generally incompatible with the individual size-extensivity of connected closed diagrams when the interaction tensor is generated by the true two-body part of the electronic Hamiltonian. Thus, in general, size-extensivity of a many-body method may originate in specific cancellations of super-extensive quantities, breaking the convenient one-to-one correspondence between the connectivity of arbitrary many-body equations and the size-extensivity of the expectation values evaluated by those equations (for example, when certain diagrams are discarded from the method). Nevertheless, assuming that many-body equations are evaluated for a stable many-particle system, it is possible to introduce a workaround, called the ε-approximation, which restores the individual size-extensivity of an arbitrary connected closed diagram, without qualitatively affecting the asymptotic behavior of the computed expectation values. No assumptions concerning the periodicity of the system and its strict electrical neutrality are made.
In this (first) paper we attempt to generalize the notion of tensor connectivity, subsequently studying how this property is affected in different tensorial operations. We show that the often implied corollary of the linked diagram theorem, namely individual size-extensivity of arbitrary connected closed diagrams, can be violated in Coulomb systems. In particular, the assumption of the existence of localized Hartree–Fock orbitals is generally incompatible with the individual size-extensivity of connected closed diagrams when the interaction tensor is generated by the true two-body part of the electronic Hamiltonian. Thus, in general, size-extensivity of a many-body method may originate in specific cancellations of super-extensive quantities, breaking the convenient one-to-one correspondence between the connectivity of arbitrary many-body equations and the size-extensivity of the expectation values evaluated by those equations (for example, when certain diagrams are discarded from the method). Nevertheless, assuming that many-body equations are evaluated for a stable many-particle system, it is possible to introduce a workaround, called the ε-approximation, which restores the individual size-extensivity of an arbitrary connected closed diagram, without qualitatively affecting the asymptotic behavior of the computed expectation values. No assumptions concerning the periodicity of the system and its strict electrical neutrality are made.
A novel reduced-scaling, general-order coupled-cluster approach is formulated by exploiting hierarchical representations of many-body tensors, combined with the recently suggested formalism of scale-adaptive tensor algebra. Inspired by the hierarchical techniques from the renormalisation group approach, H/H2-matrix algebra and fast multipole method, the computational scaling reduction in our formalism is achieved via coarsening of quantum many-body interactions at larger interaction scales, thus imposing a hierarchical structure on many-body tensors of coupled-cluster theory. In our approach, the interaction scale can be defined on any appropriate Euclidean domain (spatial domain, momentum-space domain, energy domain, etc.). We show that the hierarchically resolved many-body tensors can reduce the storage requirements to O(N), where N is the number of simulated quantum particles. Subsequently, we prove that any connected many-body diagram consisting of a finite number of arbitrary-order tensors, e.g. an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN) floating-point operations. On top of that, we suggest an additional approximation to further reduce the computational complexity of higher order coupled-cluster equations, i.e. equations involving higher than double excitations, which otherwise would introduce a large prefactor into formal O(NlogN) scaling.
The fundamentality of the exponential representation of a second-quantised correlated wave function is emphasised with an accent on the physical sense of cluster amplitudes as cumulants of the correlated ansatz. Three main wave function formalisms, namely, the configuration-interaction theory, the coupled-cluster approach, and the many-body perturbation theory (as well as their extensions, e.g. the equation-of-motion coupled-cluster method, multireference schemes, etc.), are represented in an exponential form, leading to a formulation of the working equations in terms of cluster amplitudes. By expressing the corresponding many-body tensor equations in terms of cluster amplitudes, we could unambiguously check connectivity types and the asymptotic behaviour of all tensors/scalars involved (in the formal limit of an infinite number of correlated particles). In particular, the appearance of disconnected cluster amplitudes corresponds to unphysical correlations. Besides, we demonstrate that the equation-of-motion coupled-cluster approach, as well as certain excited-state configuration-interaction methods, can be recast in a fully connected (exponential) form, thus breaking the common belief that all truncated configuration-interaction methods violate connectivity. Our work is based on the recently developed algebraic framework which can be viewed as a complement to the classical diagrammatic analysis.
Electronic structure calculations have been one of the most successful scientific fields in the past 50 years. Density functional theory (DFT) with its nice balance between accuracy and efficiency has now become the standard technique for many materials research. On the other hand, the size of systems that wavefunction-based methods can handle has also been improved thanks to the recent developments in both computation power and more advanced methodologies. However, these methods are still suffering from the curse of dimensionality. The cubic scaling of a typical DFT calculation restricts the handleable system size of DFT to only a few thousand electrons. To this end, tensor-structured techniques provide a route to constructing a reduced-order algorithm and hence are promising to improve the computational efficiency. In this dissertation work, we aim at exploring the applications of the tensor-structured techniques on different aspects of electronic structure calculations. In the first part of this work, we present a tensor-structured algorithm for efficient large-scale DFT calculations by constructing a Tucker tensor basis that is adapted to the Kohn–Sham Hamiltonian and localized in real-space. The proposed approach uses an additive separable approximation to the Kohn–Sham Hamiltonian and an L1 localization technique to generate the 1-D localized functions that constitute the Tucker tensor basis. Numerical results show that the resulting Tucker tensor basis exhibits exponential convergence in the ground-state energy with increasing Tucker rank. Further, the proposed tensor-structured algorithm demonstrated sub-quadratic scaling with system-size for both systems with and without a gap, and involving many thousands of atoms. This reduced-order scaling has also resulted in the proposed approach outperforming plane-wave DFT implementation for systems beyond 2000 electrons. In the second part of this work, we present an acceleration scheme for the proposed tensor-structured algorithm. In the proposed algorithm, the discrete Kohn-Sham problem is solved using the Chebyshev filtering subspace iteration method that relies on matrix-matrix multiplications of a sparse symmetric Hamiltonian matrix and a dense wavefunction matrix, expressed in the localized Tucker tensor basis. These matrix-matrix multiplication operations, which constitute the most computationally intensive step of the solution procedure, are GPU accelerated providing ~8-fold GPU-CPU speedup for these operations on the largest systems studied. The computational performance of the GPU accelerated code is presented using benchmark studies on aluminum nano-particles and silicon quantum dots with system sizes ranging up to ~7,000 atoms. Finally, we present a computation kernel using tensor-structured techniques to evaluate the one-electron and two-electron integrals, which are the central quantities to construct the Hamiltonian matrix of a full configuration interaction (FCI) calculation, for any molecular orbitals projected on a tensor-structured finite-element mesh. The proposed computation kernel is used to compute the integrals using Hartree-Fock molecular orbitals, which is subsequently used to perform an FCI calculation. In our numerical study, the FCI energy using integrals from the proposed computation kernel is confirmed to be consistent with a standard FCI calculation using the same Hartree-Fock orbitals. The proposed computation kernel provides a useful tool to investigate the behavior of Kohn-Sham orbitals as a basis for FCI calculations in the future, which could be a route to construct a reduced-order basis for an FCI calculation to reduce the computational cost and enable FCI calculation for the previously computationally inaccessible systems.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
Projection operators may be used instead of vectors to represent the states of a quantum-mechanical system and the appropriate equation of motion is obtained. When a basis is used the basic vectors may be oblique or orthogonal. A formulation appropriate to an oblique basis is developed with the help of tensor methods and the summation convention. This is generalized to be applicable to systems containing N electrons subject to Pauli's exclusion principle. The projection operator corresponding to an arbitrary state is the density operator and is represented by an Nth-order tensor. The well-known tensor operation of contraction of indices applied to the density operator furnishes similar, reduced operators giving one-, two-, or more-particle properties of the system. The one-particle properties are of special interest in chemical systems, the corresponding reduced density matrix being called the charge and bond order matrix by L\"owdin. The full formulas for the reduction of the general density matrix are given, showing the origin of the Coulomb and exchange operators.When the symbolic equation of motion is expressed by means of components and the process of contraction (reduction) is applied, the equation of motion of the one-particle reduction is obtained as well as an expression for the energy of the system. In the special case of the Hartree-Fock approximation the time-dependent wave equations for the one-particle orbitals are shown to satisfy the general equation of motion, suggesting that the energies of the individual orbitals need not be stationary. This approximation is characterized by the idempotent property of the one-particle reduced density matrix. This condition alone leads to a close restriction on the possible values of the charge and bond orders of a molecule. Specific formulas for the molecule butadiene show that these can be expressed in terms of two parameters only, in this case.
Abstract In this paper, the tensor-structured numerical evaluation of the Coulomb and exchange operators in the Hartree-Fock equation is supplemented by the usage of recent quantized-TT (QTT) formats. It leads to O(log n) complexity at computationally extensive stages in the rank-structured calculation with the respective 3D Hartree and exchange potentials discretized on large n×n×n Cartesian grids. The numerical examples for some volumetric organic molecules confirm that the QTT ranks of these potentials are nearly independent of the one-dimension grid size n. Thus, paradoxically, the complexity of the grid-based evaluation of the Coulumb and exchange matrices becomes almost independent of the grid size, being regulated only by the structure of a molecular system. As a result, the grid approximation of the Hartree-Fock equation allows to gain the high resolution with a guaranteed accuracy.